Currently, the dryer door is attached but not the front panel (so I can see the igniter, heat cone), but the symptoms are the same with the front panel installed.

The clue to me is the clicking sound that is present when the flame is on. And this happens at the same intervals with or without the front panel. The clicking sounds like a quick (2 times per sec) rap-rap-rap...., but only when the flame has been ignited, not when just the igniter is energized.